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Theorem eujust 1950
Description: A soundness justification theorem for df-eu 1951, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
eujust (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eujust
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1646 . . . . 5 (𝑦 = 𝑤 → (𝑥 = 𝑦𝑥 = 𝑤))
21bibi2d 230 . . . 4 (𝑦 = 𝑤 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑤)))
32albidv 1752 . . 3 (𝑦 = 𝑤 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑤)))
43cbvexv 1843 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
5 equequ2 1646 . . . . 5 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
65bibi2d 230 . . . 4 (𝑤 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑧)))
76albidv 1752 . . 3 (𝑤 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
87cbvexv 1843 . 2 (∃𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
94, 8bitri 182 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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